Class CBSE Class 12 Mathematics Vector Algebra Q #1434
KNOWLEDGE BASED
REMEMBER
3 Marks 2025 AISSCE(Board Exam) SA
If $\vec{a}$ and $\vec{b}$ are unit vectors inclined with each other at an angle $\theta$, then prove that $\frac{1}{2}|\vec{a}-\vec{b}|=\sin\frac{\theta}{2}$.
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Detailed Solution

Step 1: Express the magnitude squared of the difference of the vectors

We start by expressing the square of the magnitude of the difference of the two vectors $\vec{a}$ and $\vec{b}$: $$|\vec{a} - \vec{b}|^2 = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b})$$

Step 2: Expand the dot product

Expanding the dot product, we get: $$|\vec{a} - \vec{b}|^2 = \vec{a} \cdot \vec{a} - 2(\vec{a} \cdot \vec{b}) + \vec{b} \cdot \vec{b}$$

Step 3: Use the properties of unit vectors and the angle between them

Since $\vec{a}$ and $\vec{b}$ are unit vectors, $|\vec{a}| = 1$ and $|\vec{b}| = 1$. Also, $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = \cos\theta$. Therefore, $$|\vec{a} - \vec{b}|^2 = 1 - 2\cos\theta + 1 = 2 - 2\cos\theta = 2(1 - \cos\theta)$$

Step 4: Apply the trigonometric identity

Using the trigonometric identity $1 - \cos\theta = 2\sin^2\frac{\theta}{2}$, we have: $$|\vec{a} - \vec{b}|^2 = 2(2\sin^2\frac{\theta}{2}) = 4\sin^2\frac{\theta}{2}$$

Step 5: Take the square root

Taking the square root of both sides, we get: $$|\vec{a} - \vec{b}| = 2\sin\frac{\theta}{2}$$

Step 6: Divide by 2

Dividing both sides by 2, we obtain the desired result: $$\frac{1}{2}|\vec{a} - \vec{b}| = \sin\frac{\theta}{2}$$

Final Answer: $\frac{1}{2}|\vec{a}-\vec{b}|=\sin\frac{\theta}{2}$

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because the student needs to recall the properties of dot products, unit vectors, and trigonometric identities to prove the given relationship.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the relationships between vectors, their magnitudes, and the angles between them, as well as applying trigonometric identities. It involves understanding concepts rather than just recalling facts or following a procedure.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the student's understanding of vector algebra and trigonometric identities, which are core concepts covered in the textbook.

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